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4
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0answers
87 views

How to model this easy problem as sum of indicator random variables in order to apply Chernoff bound

Do you have an idea how I could model the following process somehow as a sum of independent indicator random variables? I have given a grid of size $n \times n$ for $n \rightarrow \infty$. Now I ...
3
votes
2answers
48 views

Variation of Chebsyhev: How to prove that?

I have the "job" to prove that for any random variable with standard deviation $\sigma$ and expectation $\mu$ and for any $t>0$ we have $$Pr[X-\mu \geq t \sigma] \leq \frac{1}{1+t^2}.$$ I thought ...
0
votes
1answer
19 views

Tail estimates for Binomial with constant mean

The Chernoff Bound gives a good tail estimate for a Binomial Distribution, but only if the mean goes to infinity. However, for a constant mean, Chernoff bound does not help at all. Is there some ...
4
votes
1answer
79 views

My data is not normally distributed: what can I do to estimate a tail probability?

Continuing on from my earlier question, I'm attempting to analyse the data qualitatively. In the following plot, I make $10000$ samples where I count "the number of clashes". I plot $n$ vs. the ...
0
votes
0answers
23 views

How to Prove this Multinomial Distribution Inequality

I have the following lemma, but there seems to be one (or two) mistakes in the proof found in this paper (lemma 3). The lemma states that for $Multinomial(n,p_1,\ldots,p_k)$ distributed ...
1
vote
0answers
112 views

Concentration inequality of weighted sum of random variables given a tail inequality

I tried to solve the exercise below which can be seen as a generalization of the Bernstein's concentration inequality. However, I have difficulty bounding the moment generating function of $Z$ (see ...
1
vote
1answer
24 views

limiting behavior of standard normal survivor function [duplicate]

How do you show that $\lim_{x\to \infty} 1-\Phi(x) \sim \phi(x)/x$? In the previous, I'm using $\Phi$ to refer to the standard normal CDF and $\phi$ to refer to the standard normal pdf. Thanks!!
0
votes
0answers
9 views

Is the following a proper application of Hoeffding's inequality?

I'm somewhat shaky on my probability and was wondering if someone could double-check that the following is a valid application of Hoeffding's inequality. Suppose that I have a random variable $X \sim ...
0
votes
1answer
62 views

Levy flight distribution

Can somebody help me with C++ code, how to make a Levy distribution like a function? I need to make one dimensional Levy flight model, but I don't know the function how to choose the right step.
2
votes
1answer
107 views

Is Wikipedia correct?

Cantelli's Theorem Wikipedia says: $$P[X-\mu \geq a] \leq \frac{\sigma^2}{\sigma^2+a^2}$$ for $a > 0$ and $$P[X-\mu\geq a] \geq 1- \frac{\sigma^2}{\sigma^2+a^2}$$ for $a <0$. Is the second ...
1
vote
1answer
30 views

“Reverse” distribution-tails

Chernoff, Markov and Chebyhev all give some upper bound for tail probabilities, e.g. Chebyshev gives us $Pr[|X-E[X]| \geq t] \leq \frac{Var[X]}{t^2}$. This is quite helpful, but what if I would ...
0
votes
1answer
40 views

Tail of a sequence of RV

Consider the sequence of random variables $\{X_n\}_{n=1,2,\dotsc}$ where $X_n$ is gamma-distributed with shape $n$ and scale $1/n$ (or equivalently, $2nX_n$ is $\chi^2$-distributed with $2n$ degrees ...
2
votes
1answer
57 views

Inequalities for the tail of the normal distribution (Halfin-Whitt paper)

I am reading the famous paper by Halfin and Whitt, [1]. I'd like to prove remark (1) on page 575. The authors state \begin{align} \frac{\beta \alpha}{(1-\alpha)} = \frac{\phi(\beta)}{\Phi(\beta)} ...
1
vote
1answer
113 views

Tail bound sum of unbounded RVs of given mean and variance?

Let $X_1,\dots,X_n$ be independent variables, $X_i$ having mean $\mu_i$ and variance $\sigma_i^2$. Let their sum $S = \sum_{i=1}^n X_i$. Of course, $S$ has mean $\lambda = \sum_{i=1}^n \mu_i$ and ...
2
votes
1answer
75 views

Mnemonic for platykurtic and leptokurtic

I keep confusing terms leptokurtic and platykurtic. Is there a good mnemonic to help remember which is which? "Lepto" means "little", "platy" means "flat", and both are equally unrelated to thickness ...
1
vote
0answers
99 views

Long tail distributions

I have just started to study sub-exponential distributions. I wonder how to prove this statement : Let $X_1,\dots,X_n$ are independent random variables with distributions $F_1,\dots,F_n$ respectively. ...
1
vote
1answer
285 views

difference between upper and lower tails of Chernoff bounds

I'm struggling with the intuition behind why Chernoff bounds differ in the upper and lower tails. That is, for the lower tail we have: $$ Pr(X \le (1 - \delta)\mu)\ \ \le\ \ ...
2
votes
1answer
77 views

Conditions on r.v. $X$ s.t. $\Pr(X\ge n\mid X\ge n/2)$ gets small?

So I have a random variable $X$ that takes values in the non-negative integers. I want it to have one of the following properties: either $$\lim_{n\to\infty} \Pr(X\ge n\mid X\ge n/2)=0$$ or there ...
6
votes
3answers
268 views

Proof that $E(X)<\infty$ entails $\lim_{n\to\infty}n\Pr(X\ge n) = 0$?

As the title says. I think this should follow straightforwardly but I can't find a proof. My random variable of interest $X$ takes values in the non-negative integers. The only other assumption on ...
1
vote
1answer
55 views

What is the null hypothesis for a 1-tailed test?

I've been given different answers to this question in different courses. Some professors say it is (using the example alternative hypothesis of $\mu > 3$): $$H_0: \mu = 3$$ $$H_1: \mu > 3$$ ...
2
votes
1answer
171 views

'Simple" Chernoff bounds

Reading an academic paper I've observed the next claim: A simple Chernoff argument will now show that if an event has a constant probability at every step of occurring and there's independence ...
1
vote
1answer
112 views

Tail bound for hypergeometric distribution

I am looking for a reference (book) for the tail bound for the Hypergeometric distribution. I know there is a nice paper by Skala (2009) but its unpublished. I am looking for a book which would be a ...
1
vote
2answers
198 views

Coins and probability

Bob has $n$ coins, each of which falls heads with the probability $p$. In the first round Bob tosses all coins, in the second round Bob tosses only those coins which fell heads in the first round. Let ...
1
vote
1answer
173 views

Chernoff bound for Geometric RVs compared to exact tail bound

I keep getting a result I can't interpret. X is a Geometric RV with distribution ($0<\rho<1$) $$ \pi_k = \rho^k(1- \rho) $$ so directly applying Geometric series the tail bound is $$ B_1 = ...
1
vote
1answer
308 views

Fat Tail / Large Kurtosis Discrete Distributions?

All, I'm wondering if there are any notable, basic discrete probability distributions with "fat/heavy tails" or a large kurtosis? I know the Geometric Distribution's excess kurtosis approaches 6, ...
0
votes
1answer
262 views

Upper-tail inequality for t-distribution

I am interested in upper tail bounds (or bounds on deviation from the mean) for t-distribution with n degrees of freedom (http://en.wikipedia.org/wiki/Student's_t-distribution) A bound that is of the ...
5
votes
2answers
418 views

Repeatedly rolling a die and the tails of the multinomial distribution.

For $1\leq i\leq n$ let $X_i$ be independent random variables, and let each $X_i$ be the uniform distribution on the set ${0,1,2,\dots,m}$ so that $X_i$ is like an $m+1$ sided die. Let ...